# How to teach calculus (book recommendation)

I'm going to teach calculus for the first time to undergraduate students. I would like to know if there is some book about how to teach the concepts of calculus (e.g. limits, derivatives, etc.).

• I don't have a book to recommend, but I would be happy to chat with you about it, and share resources. – Sue VanHattum Jul 31 '15 at 21:30
• Look at all the calculus textbooks you can get your hands on. Some of them will have ways of explaining things that you've never seen before that will be instructive. – Alexander Woo Jul 31 '15 at 23:35
• I didn't mean buying any books. Many of your more experienced colleagues wherever you are will have a half dozen different calculus textbooks on their shelves and be quite happy to lend them to you. – Alexander Woo Aug 1 '15 at 1:16
• I'm going to check out some of these suggestions myself. Thanks, all. user26832, you might want to make a profile with a name, so people will know who you are. You can email me at mathanthologyeditor @ gmail. – Sue VanHattum Aug 1 '15 at 21:33
• Joseph, what do you admire about his teaching? (What makes him a master teacher?) – Sue VanHattum Aug 1 '15 at 23:55

There are so many Calc 1, 2, and Multi video lectures online now. Watch a few... take notes on what you think are some good traits of the person teaching. MIT's Calc course (super fast) might be helpful if you're teaching that style and speed.

Some resources

1. This text, How to Teach Mathematics - S. Krantz, has many useful guides for the beginning person. I don't agree with all, but it's a great resource.
2. Suzanne Kelton wrote a guide for the AMS. Its pdf is here: teaching Mathematics at the college level - pg. 31 goes into Calc stuff!
3. Also, do not be ashamed to search for AP Calculus teachers' resources. It may be a "high school" course, but some of the teachers out there are amazing and offer a lot of guidance. Specific topic choice may vary but good teachers are that no matter what. Lin McCullin's website is an prime example. Watch those videos yourself from his site :) Ignore test-specific things. Pedagogy!
4. If you are in the US, go to your MAA sectional meeting in the Fall!
5. Please, please, please incorporate technology as a reasonable tool that you actually teach how to use in the right situations. This is vital and super valuable.

Numbers and Functions: Steps into Analysis, R.P. Burn, Cambridge University Press, ISBN 0 521 78836 6.

This book takes a constructivist approach to the subject. It takes the form of a sequence of problems, introducing the key ideas step by step.

In his preface Burn identifies common problems student often encounter: the apparently needlessly formal approach of topics that are familiar from school as well as the insistent use of arbitrary and contrived definitions. Accordingly the book builds on high school concepts and naturally introduces key definitions.

Personally it saved me in my first year of uni.

Below is a review from 'The Mathematical Gazette'

http://www.jstor.org/stable/3621803?seq=1#page_scan_tab_contents

Mathematics should essentially be fun to do and to read and I think that generally students only love to do those things which interests them. So, one basic (and I think mandatory) component of your teaching should be about making the subject interesting for the students.

Presuming that by Calculus you mean Real Analysis I have the following suggestions for you.

• First, let the students feel themselves the importance of a rigorous study of calculus. For this, I think that the introductory chapters of Tom M. Apostol's Calculus I and Terence Tao's Analysis I are two excellent references. Also I would like to recommend reading the last three sections of Part II of the book The Princeton Companion to Mathematics.

• Second, when you will be giving assignments to your students try to be scientific in choosing the problems. To be precise, I don't recommend giving difficult (it's a relative term, though) problems to the beginners, nor do I recommend giving a set of problems which doesn't have any relation(s) among them. Though, it's difficult to express precisely what I mean by the above statements, I will nevertheless try to give an example which is intended to make the matter a bit more clear.

Consider the following problems,

1. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $f(x)=0$ for all $x\in\mathbb{Q}$. Show that, $f(x)=0$ for all $x\in\mathbb{R}$.

2. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Let $T=\left\{\dfrac{m}{2^n}\mid m,n\in\mathbb{Z}\right\}$ such that we have $f(x)=0$ for all $x\in T$. Can we say that $f(x)=0$ for all $x\in\mathbb{R}$?

3. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Let $S=\left\{\dfrac{a}{2^n}\mid a, n\in\mathbb{Z}\right\}$ ($a$ is a constant) such that we have $f(x)=0$ for all $x\in S$. Can we say that $f(x)=0$ for all $x\in\mathbb{R}$?

These problems have a very curious property. To solve the first problem, you need to show that $\mathbb{Q}$ is dense in $\mathbb{R}$. The second problem you need to show that a certain subset of $\mathbb{Q}$ is dense in $\mathbb{R}$. Finally, the third problem asks you to prove that a subset of the subset of Problem 2 is dense in $\mathbb{R}$. So you can see that the first problem is follows from the second one, whereas the second problem follows from the third one. Can you appreciate the order in which the exercises are arranged ? Can you intuitively grasp what I meant in the above lines ?

• Third, make your students familiar with some basic ideas of Mathematical Logic at the very beginning. For example, the concept of conjunction, disjunction, negation etc.

Finally, I think that for teaching calculus L. V. Tarasov's book Calculus: Basic Concepts for High Schools is an invaluable guide.

• Calculus = Real Analysis is a foolish assumption to make in America,sadly. These are all good recommendations for an honors course at a typical university,but not much else. – The Mathemagician Feb 25 '18 at 19:34